Independence from Similar
Alternativesin Social Choice Procedures

Stephen Eppley <seppley@alumni.caltech.edu>

Revised: January 29, 2003

Abstract

Unless the set of alternatives is
fixed by nature, election outcomes may be manipulated by
adding alternatives to (or
removing alternatives from) the set of nominees. Satisfaction
of TN Tideman's criterion independence of clone alternatives (ICA) may be more
important
than satisfaction of other criteria such as reinforcement, participation
and cancellation since voting procedures which fail ICA appear to be much more manipulable
than those which satisfy ICA. (The Borda procedure satisfies reinforcement,
participation
and cancellation but fails ICA and even fails a weaker criterion
independence from exact
clone alternatives.)

The existence of Condorcet-consistent
voting procedures which also satisfy a stronger
criterion independence from similar alternatives (ISA) is demonstrated, indicating
resistance to clone manipulation is robust under voter preference perturbations which
make ICA's "clone" premise not strictly hold. Furthermore, it is established in
other
documents that voting procedures can satisfy not only ICA and ISA but also other
desirable
criteria such as top cycle, monotonicity,
resolvability,
strong Pareto, minimal
defense, truncation
resistance and immunity
from majority complaints.

1. Introduction

Considerable social choice
literature exists regarding manipulability of voting procedures
by voters misrepresenting their preferences. But this is not
the only kind of strategy by
which outcomes can be manipulated. If the set of
alternatives is endogenous (not fixed by
nature), outcomes can be manipulated by adding
alternatives to or removing alternatives
from the set of alternatives being voted upon. Such strategies may take various
forms:
One
is to add an alternative which reveals voters' collective preferences are cyclic
(Condorcet's paradox), with the hope or expectation that the procedure will resolve
the cycle favorably from the point of view of the manipulator. However,
this particular
strategy may sometimes be deemed benign, when there is no a priori reason to
believe
the additional preference information leads to an outcome worse
for the society.

Another such strategy requiring little
skill is to add (or remove) alternatives which are
"similar" to another already on the ballot. Institutional rules often make it easy for
individuals or for small minorities to add alternatives to the ballot; thus this
form of
manipulation may be easier
than coordinating a coalition to strategically misrepresent
preferences or gerrymandering of subsets of voters. A lone
individual can remove
an alternative (him/herself) when the context is electing
a person to an office, and
in some societies a lone individual can add
him/herself to the set of alternatives. In
many assemblies only two
members are needed to add an alternative, one to propose
and another to second the proposal. Since it is easy to generate
similar proposals
and since it may be fairly easy to find additional candidates who espouse policies
similar to those advocated by other candidates, and since a would-be manipulator
can
reasonably expect voters to be nearly indifferent between similar alternatives,
manipulation using similar alternatives may
occur frequently if the voting procedure
is not independent from similar alternatives.

A familiar
example of this strategy is when a political party nominates only one
candidate to run for an office even though they are divided over which party
member
is the best for the office (or divided over who has the best chance of winning),
when
the voting procedure in the general election is plurality rule, majority runoff,
Hare
(also known as Instant Runoff, Single Transferable Vote and the Alternative
Vote) etc.
The party's winnowing process (e.g., partisan primary elections) may itself be flawed--
sensitive to financial
resources, voter turnout, etc.--and manipulable. Thus reasonable
candidates can be deterred from competing for office, out of fear
that a similar candidate
who would otherwise win will be defeated and
a candidate who is worse elected.
(That outcome is commonly known as a "spoiled"
election.) As an example, consider
a hypothetical election
with three potential candidates: Gore and Bradley (Democrats)
and Bush (Republican). Suppose the Democrats believe
both Gore and Bradley would
beat Bush if either one of them ran against
Bush. In other words, they believe Gore and
Bradley are similar candidates according to our
definition in section 2. Under plurality
rule, however, the Democrats can expect the voters who prefer Gore and Bradley
over
Bush would split their votes and Bush would be elected, so they would nominate
only
one of them to run against Bush. This strategy may deprive the voters the
chance to
elect the best candidate. But given a voting procedure which is independent from
similar candidates, it would be unnecessary to nominate only one of them.
Indeed,
they would have incentives to nominate more than one: to increase the turnout of
the
party's supporters on election day, and to avoid putting all their eggs in one
basket.

For an example involving the Borda
voting procedure (which awards a point to
each
candidate for every candidate it
outranks, for each voter, then elects the candidate having
the most
points), consider a
spatial model involving two salient issues in which three
candidates x, y and z take
positions on the issues, and y's positions are similar to z's:

yzx

Suppose there are five voters,
each having preferences
regarding the possible positions.
Suppose x is at the ideal point of voters 1-3 and y is at the
ideal point of voters 4-5.
If only x and y compete, one may predict x would be chosen
under any voting procedure
since a majority prefer x over y. And since it is difficult,
if not impossible, to compare
intensities of voters' preferences, we have no reason to believe voters 4-5's preferences
for y over x should be accorded more weight than 1-3's preferences for
x over y. Thus
it is not only predictable that x would be chosen if only x and y
compete, it is reasonable
to choose x if only x and y compete. Suppose also
that voters 1-3 rank y over z since y is
slightly closer than z to their ideal. Thus voters 1-3 rank x
over y over z. And suppose
voters 4-5 rank z over x
since z
is much closer than x to their ideal. Thus voters 4-5 rank y over z over x. The Borda procedure erroneously divines
stronger preferences by 4-5
for y over x than by 1-3 for
x
over y since 4-5 rank z between y and x, and thus
would
elect y. Thus Borda is
prone to manipulation by adding or removing similar alternatives
such as z to the
ballot.

IECA, ICA and
ISA (defined in section 3) are
weaker criteria than the modern form of
Kenneth Arrow's independence of irrelevant alternatives (IIA),
which requires the
outcome not change if a losing alternative is deleted from the voters' rankings.
That is
such a strong criterion that no reasonably democratic voting procedure which
tallies
rankings can always comply. IECA, ICA and ISA demand voting
procedures satisfy a
form of consistency having to do
with alternatives which are sufficiently similar to other
alternatives
that adding them to the ballot (or removing them) should not significantly
affect the outcome. If the outcome can be affected by adding or
removing similar
alternatives, the outcome may be easily manipulated in this manner, particularly if
voters are
unsophisticated or have their strategic options constrained by
accountability
to constituents who are unsophisticated.

We argue that since manipulation
of procedures which fail these criteria is so easy, they
are more important than the reinforcement criterion [Young,
1974], the participation
criterion [Moulin,
1988], and cancellation. Reinforcement requires that when the votes
of two separate groups of voters would each elect the same alternative, their votes
aggregated together must also elect that alternative. (Manipulation based on
failure
of reinforcement can be considered a special case of gerrymandering.)
But since
institutional rules can easily deny to a minority the power to choose between an at-large
vote and a
strategically-designed division of the voters into subsets, it can be made
impossible for a minority to exploit failure of reinforcement. (Another option, if a society
is concerned about manipulation due to failure of reinforcement, is to modify their voting
procedure so that whenever a voter subset's top cycle
includes the alternative which
would win the at-large tally, that at-large
winner is the choice of the subset.) Hence we
argue that satisfaction of reinforcement is of little importance for deterring manipulation.
As Hervé Moulin [1988] wrote, reinforcement is reasonable "when the overall voting
body is likely to
split into many subsets." We consider it unimportant when such
splitting
cannot occur or must occur or cannot be easily exploited by a minority.

Another argument for reinforcement
is its aesthetic appeal to our sense of rationality.
In other words, it is a consistency axiom. But IECA has the same appeal since voters'
indifference between exact clone alternatives provides no
additional information which
should affect the outcome. Thus IECA is also a
fundamental consistency axiom. To a
lesser extent, ICA and ISA also share that appeal to consistency.

We also argue that IECA, ICA and
ISA are
more important than participation
[Moulin,
1988]. Participation requires no voter must ever prefer the outcome
given her abstention (non-participation) more than the outcome given her sincere vote.
But when a voter has information about other
voters' preferences and strategies which
suggests she would likely prefer the outcome
given abstention over the outcome given
her sincere vote,
that information should suffice to suggest a voting strategy better than
abstention. Thus it seems myopic to focus on abstention versus sincere voting
and
ignore other strategies, particularly strategies where the only misrepresentation
is
expression of indifference between a compromise and a preferred
alternative.
Our argument
against requiring satisfaction of participation may be less persuasive
when considering unsophisticated voters, but they are also the most
vulnerable to
being manipulated by addition or deletion of similar alternatives. There is also
the
relevant empirical question regarding whether a significant number of voters would
choose not to participate simply due to general knowledge that the
voting procedure
does not always satisfy participation; the answer may well be that insignificant numbers
of voters abstain due to this occasional failure and that other factors,
such as other
flaws in the voting procedure would be much more significant.
For both these reasons,
it seems premature to assert without empirical evidence that participation is as
important
as criteria such as IECA, ICA and ISA.

Here are three examples which demonstrate violations of
IECA, ICA and ISA by the
Borda procedure. For greater clarity, the first and second examples assume voters
are
allowed to vote indifference. Those examples could be easily modified,
however, to
demonstrate Borda violations even when votes must be strict rankings of the
alternatives
(except of course that technically no alternatives are called exact clones if
voters cannot
express indifference).

In example 1.1, x and y can be considered exact
clones since every voter is indifferent
between them. Borda
elects w with (7×3)+(4×0)=21
points, since x and y
have only
(7×1.5)+(4×2.5)=20.5
points and z has even fewer (4). But Borda would elect x if y
were not on the
ballot (and would elect y if x were not on the ballot) since (7×1)+(4×2)
exceeds (7×2)+(4×0).
In a scenario where x and y are similar to each other but not
exact clones, the better of the two might be deterred from competing, implying a loss
for society.

In example 1.2, y and z can be considered exact
clones since every voter is indifferent
between them. Note y and z are ranked last by most voters
and next to last by the other
voters. Thus it seems reasonable to say neither y nor z
should be elected (and under
most voting procedures their presence on the ballot would not affect the
outcome).
Borda elects x with (5×2)+(2×3)=16
points, since w has
only (5×3)+(2×0)=15 points
and y & z have only (5×0.5)+(2×1.5)=5.5
points. But Borda would elect w if z were
not on the ballot since (5×2)+(2×0)
exceeds (5×1)+(2×2).
This implies Borda outcomes
can be affected by the presence on the ballot of sure-losers
("turkeys" in Gary Cox'
terminology).

In example 1.3, y and z are
deemed clones since no voter ranks any alternative between
them. (That is, every voter who rank x over y also ranks
x over y', every voter who ranks x over y'
also ranks x over y, every
voter who ranks y over x also ranks y' over x, and
every voter who ranks y' over x also
ranks
y over x.) Borda elects y with (3×1)+(2×2)=7
points, since x
has only 3×2=6 points and y' has only 2×1=2
points. But
x would win if y' were not on the ballot (3>2). Clearly the "y &
y'" minority party has an incentive to
place both y and y' (and as many clones of y as possible) onto the ballot.
(Note that
this is like the problem illustrated above in the two-issue spatial model.)
Similarly, the
"x" party has an incentive to place as many clones of x
onto the ballot as possible,
so elections under Borda could become farces.

Examples showing the Copeland procedure fails IECA,
ICA and
ISA and the Simpson-
Kramer procedure fails ICA and ISA can easily be
constructed.

ICA is essentially equivalent to a criterion promoted by
TN Tideman, TM
Zavist,
and
others. IECA is weaker and ISA is
stronger. IECA is the easiest of the three criteria
to justify since the massive indifference between exact clone alternatives means the
voters are not really providing any additional information about their preferences
which
should
reasonably affect the outcome. ICA is nearly as easy to justify, differing from IECA only in that voters need not be
indifferent between alternatives as long as they
do not rank other alternatives between them; this is a strong indicator
that these
"clone" alternatives are indeed similar to each other and thus the
voters' preference
intensities between clones are likely milder than between clone and
non-clone.

ISA is much broader than ICA, and
variants of ISA which cannot together be satisfied
could be plausibly defined. These considerations raise the
question whether ISA is
overly broad, which is not answered
here. However, even if ISA is deemed too
strong to require, voting procedures which satisfy ISA demonstrate
their satisfaction
of ICA is robust. In practice ICA's premise would rarely strictly hold since with many
voters there could easily be a few
odd voters who rank dissimilar alternatives between
similar ones, and thus robust satisfaction of ICA is important when
evaluating the ease
of manipulability.

2. The Framework

First some notational conventions. See the
document "Set
Operators and Binary Relations"
for an explanation of the mathematical symbols used below. (Note that some
symbols may
not display properly when viewed with a browser which does not
support HTML 4.0 or
if the
browser is not configured to display the "Symbol" font.)

Let V denote a finite non-empty set of voters {1,2,3,...,n}.
Let A denote a non-empty set of
potential alternatives.
Let B, a finite non-empty subset of A, denote the
alternatives nominated to the ballot.

The set of nominees B is determined endogenously under some unspecified
procedure.

For all non-empty Z ⊆ A, let
Rankings(Z)
denote the set of all possible
non-strict rankings (also called weak orderings) of the alternatives in Z.

Each voter is assumed to have transitive asymmetric preferences regarding the alternatives
in A. Each voter casts a ballot by expressing a ranking of B.
For all i ∈
V, let Ri denote
the ballot cast by voter i. Thus Ri
∈ Rankings(B) for all i ∈
V. (We have not assumed
voters express strict rankings (linear orderings) because, even if it is
reasonable to assume
every voter sincerely has strict preferences, there may be
strategic incentives to report
indifference, and there may be unsophisticated
failure to report all sincere strict preferences
since a large set of nominees would make that tedious. Furthermore, it is not a priori
desirable to prevent the voters from voting indifference strategically, since some strategies
involving
insincere indifference may deter or thwart other voters' socially harmful strategies.
See the document
discussing the minimal defense criterion. Also, requiring
each vote
to be a strict ranking could deter voters from voting if the set of nominees is
large.)

Let R denote the set of voters' ballots. For all a,b
∈ B, let R(a,b) denote
the
ballots in R which rank a over b. Thus #R(a,b)
is the number of voters who
ranked a over b.

For all finite non-empty B ⊆ A
and all C ⊆ B, C is a set of exact clone
alternatives within B given R if and only if both of the following conditions
hold:

(Condition 1 says every voter
expresses indifference between exact clones. Condition 2 is
an innocuous technical condition which implies an exact clone
set is not contained in a larger
exact clone set. This avoids
a problem in some rare scenarios involving both pairwise ties
and massive indifference. It
refers to an alternative which, though technically might be
considered outside an exact
clone set, is itself an exact clone of those in the exact clone
set. Would-be manipulators would consider such an alternative an
exact clone too, and
we are only interested in ensuring their manipulation of the nominations of an exact clone
set cannot affect the election
of alternatives they would not consider exact clones. Thus
condition 2 is innocuous, and means that a strict subset of a set of exact
clones is not
called a set of exact clones.)

For all finite non-empty B ⊆ A
and all C ⊆ B, C is a set of clone alternatives
within B given R if and only if both of the following conditions
hold:

(Condition 1 says every voter who
ranks a clone over a non-clone ranks every clone over
that non-clone, and every voter who ranks a non-clone over a clone ranks
that non-clone
over every clone. Condition 2 is
an innocuous technical condition which implies C is not
contained in a larger set of exact clones,
which permits condition 1 to be slightly relaxed
compared to Tideman's definition of clones.)

Since we are discussing the effect on outcomes when
the set of nominated alternatives B
is changed, the notation must be able to indicate deletion of alternatives from B
and R:

For all non-empty Z ⊆
A and all i ∈ V, let RiZ
denote the restriction of Ri to Z.
In other words, RiZ is the same as Ri
except all alternatives not in Z are deleted.
Thus RiZ ∈ Rankings(Z)
for i ∈ V. Let RZ denote the
restriction of R to Z.
That is, RZ = {R1Z,R2Z,...,RnZ}.

We are primarily interested in voting procedures
which leave little or nothing to chance.
(Completely eliminating chance
would require sacrificing anonymity or neutrality,
desirable
democratic properties which require no voters nor alternatives be
specially
privileged.) However, we use a framework general enough to encompass both
deterministic and probabilistic voting procedures.

A lottery
is a #A-tuple of real numbers such that each component corresponds to
an
alternative in A, each component is a real number in [0,1] and the
sum
of the
components is 1. For all lotteries L and all x ∈
A, let Lx denote the component
of L which corresponds to x. Thus, both of the following
conditions hold for
all lotteries L:

Lx is interpreted as the probability
x will be
elected. Thus, Lx = 1 implies x will be
elected with certainty, and Lx = 0 implies x will be
defeated with certainty.

A voting scheme is
a function f which maps a non-empty Z ⊆
A and a set of
voters' rankings R into a lottery and satisfies the following conditions:

(1) fx(Z,R)
= 0 for all A, all R, all non-empty Z ⊆ A
and all x ∈ A\Z.
(2) fx(Z,RZ)
= fx(Z,R) for all A, all R, all
non-empty Z ⊆ A, and all x ∈
A.

Just as Lx denotes the
component of lottery L which corresponds to alternative x, fx
denotes the component of f which corresponds to x. If fx
= 1, x will be elected with
certainty. If fx = 0, x will be defeated with
certainty. If 0 < fx < 1, it is presumed
some
fair random mechanism will determine the outcome in accordance with the
probabilities.
A voting scheme f is deterministic if and only if fx(Z,R) is 1 or 0 for all A, all R,
all non-empty Z
⊆
A
and all x ∈ A. A voting scheme is probabilistic
if and only if
it is not deterministic.

Condition 1 of the definition of
voting scheme means no
alternative outside Z will be
elected. Since we are discussing the effect of attempts to
manipulate by adding or
removing alternatives, the notation needed to
encompass the concept of neglect of
alternatives so outcomes given any subsets
of A
can be compared. This is why the
definition of voting scheme includes as a parameter
the subset of alternatives which can
be elected, which (after Kenneth Arrow) we call the feasible
subset. Condition 2 is
a mild restriction which
simplifies the notation by requiring the lottery must not depend
on
expressed preferences regarding alternatives outside the feasible subset. This
avoids potential complications which may otherwise
arise if voters rank alternatives
which are not in the feasible subset and
allows us to more easily model attempts to
manipulate outcomes by manipulating nominations.

Independence from Exact Clone Alternatives(IECA):
A voting scheme f is independent from
exact clone
alternatives if and only if
fx(Z,R) = fx(Z\C0,RZ\C0) for all A, all R, all
finite non-empty Z
⊆
A,
all C which is a set of exact clone alternatives within Z given R, all x ∈ Z\C
and all C0
which is a strict subset of C.

IECA demands that the probability of
election of an alternative must
be unaffected by
addition or removal of exact clones, as long as the alternative is
not itself one of the exact
clones and at least one of the exact clones is nominated. Preferences
regarding additional
exact clones must be irrelevant to the outcome. This
can be viewed as a fundamental
consistency axiom. Note that since the sum of probabilities over all
alternatives must
total 1, satisfaction of IECA also implies Σ(c ∈
C)
fc(Z,R) = Σ(c
∈ C)
fc(Z\C0,RZ\C0).

Independence from Clone Alternatives(ICA):
A voting scheme f is independent from clone
alternatives if and only if
fx(Z,R) = fx(Z\C0,RZ\C0) for all A, all R, all
finite non-empty Z
⊆
A,
all C which is a set of clone alternatives within Z given R, all x ∈ Z\C
and all C0
which is a strict subset of C.

ICA demands that as long as at least one clone is nominated,
additional clones must be
irrelevant to the outcome, except
possibly to elect a "better" clone instead of a "worse"
clone. Note that since the sum of probabilities over all
alternatives must total 1,
satisfaction of ICA also implies Σ(c ∈
C)
fc(Z,R) = Σ(c
∈ C)
fc(Z\C0,RZ\C0).

Clearly any set of exact clone alternatives is also a set of clone
alternatives. Therefore,
satisfaction of ICA implies satisfaction of IECA, and failure of
IECA implies failure of ICA.

Turning now to the definition of "similar" (near-clone) alternatives, an intuitive way to
relax the
definition of clones is to only require that in every pairing of a similar
alternative
versus an alternative outside the similar set, the "majority
preference" relation must
remain invariant when any similar alternative is substituted
for the one in the pairing:

Let Pairs denote the set of
all possible ordered pairs of alternatives in A. That
is, Pairs = {(a,b) such that a,b ∈
A}. (Note that (a,b) and (b,a) are distinct pairs.)

Let Majorities(R)
denote the subset of ordered pairs {(a,b) ∈ Pairs
such
that #R(a,b) > #R(b,a)}.
Let M denote the binary relation on A such that,
for all a,b ∈
A, aMb if and only if (a,b) ∈
Majorities(R).

For all finite non-empty B ⊆
A and all C ⊆ B, C
is a set of similar alternatives
within B given R only if both of the following
conditions hold for all c,c' ∈ C
and all x ∈
B\C:

(1) (x,c) ∈
Majorities(R) if and only if (x,c')
∈ Majorities(R).
(2) (c,x) ∈ Majorities(R)
if and only if (c',x) ∈ Majorities(R).

Note the "only if" construction instead of the usual "if and
only if." Using "if and only if"
would produce such a broad definition that it would be impossible for any
reasonable
voting scheme to satisfy independence. Thus an additional restriction involving other
aggregate
properties needs to be imposed.

Given A, R and
finite non-empty Z ⊆
A, and abbreviating k = #Z, call
the sequence z1,z2,...,zk
a majority cyclic permutation of Z given R
if and only if Z = {z1,z2,...,zk}
and (zk,z1) ∈
Majorities(R)
and (zj,zj+1) ∈
Majorities(R) for all j ∈
{1,2,...,k-1}.

Given A,
R, and finite non-empty Z ⊆ A, let
majoritycycles(Z,R)
denote
the set of subsets of alternatives {MC such that MC is a non-empty
subset
of Z and there exists a majority cyclic permutation of MC given R}.

Given A, R,
and finite non-empty Z ⊆
A, let majorities(Z,R) denote the
set of ordered pairs of
alternatives {(x,y) ∈ Pairs
such that there exists a
majority cyclic permutation of Z given R
in which x immediately precedes y}.

Given A, R,
and non-empty Z ⊆
A,
let min(Z,R) denote the set of ordered
pairs of
alternatives {(x,y) ∈ Pairs
such that (x,y) ∈ majorities(Z,R)
and
#R(x,y) <
#R(z,w) for all (z,w) ∈
majorities(Z,R)}.

When R is
clear from the context, we abbreviate majoritycycles(Z),
majorities(Z)
and min(Z).

Note that majorities(Z) and
min(Z) do not
include all "majorities" which are identifiable
in Z. Only those pairs of alternatives which are adjacent in a majority cyclic permutation
of Z are contained in majorities(Z) and
min(Z).
Two examples illustrate this:

(2.11.1) Given the relations aMb, bMc,
and c~a, majorities({a,b,c}) = Ø
and min({a,b,c}) = Ø because there are no
majority cyclic permutations of {a,b,c}.

(2.11.2) Given the relations aMb, bMc,
cMd, dMa, aMc, and dMb, majorities({a,b,c,d})
= {(a,b),(b,c),(c,d),(d,a)}.
Without more information, min({a,b,c,d})
cannot be specified, but it is always the case that min(Z,R) ⊆
majorities(Z,R). Note that (a,c) and (d,b)
are not in majorities({a,b,c,d}) and are
not in min({a,b,c,d}), even though aMc
and dMb, because there are no majority cyclic permutations of {a,b,c,d}
in which a immediately precedes c or in which d
immediately precedes b. For instance (trying to find a
permutation in which a immediately precedes c), acdabcd
is not a majority cyclic permutation of {a,b,c,d}because
its alternatives are not all distinct, and acd is not a majority cyclic
permutation of {a,b,c,d} because it does not
include all four alternatives in {a,b,c,d}. However,
even though (a,c) ∉ majorities({a,b,c,d}),
(a,c) ∈
majorities({a,c,d}).

The following two definitions are not technically necessary for a definition
of an ISA criterion
which is stronger than ICA, but will be used to broaden
similarity in order to strengthen ISA
even further.

Definition (2.12): Given A, R
and non-empty Z ⊆ A, and abbreviating k =
#Z, the sequence x1,x2,...,xk is a cyclic permutation
of (Z,R) if and only if x1, x2, ..., xk
are distinct alternatives in Z and [xkMx1 or xk~x1]
and [xjMxj+1 or xj~xj+1 for
all j ∈ {1,2,...k-1}]}. Given A, R, and non-empty Z ⊆
A, let cycles(Z,R) denote the set of subsets of
alternatives {C such that C is a non-empty
subset of Z and there exists a cyclic permutation of (C,R)}.

Definition (2.13): Let τ(Z,R)
denote the top cycle of (Z,R). Formally, given A, R,
and non-empty Z ⊆ A, τ(Z,R) is the set
of alternatives {x ∈ Z such that [for all z ∈
Z such that zMx, there exists a C ∈ cycles(Z,R)
such that x,z ∈ C]}.

When R is
clear from the context, we abbreviate cycles(Z) and τ(Z).

Definition (2.14): For
all x,y ∈ A
and any set S which is either a set of alternatives or a set of ordered pairs
of alternatives such that y is nowhere in S, let substitute(S,x,y)
denote the set obtained by substituting y for x everywhere x appears in S.

Definition (2.15): A set Y is
a similar set of (Z,R) if and only if the following three conditions
hold:

(2.15.1) Y ⊆ Z ⊆
A and #Y > 1.

(2.15.2) For all y,y' ∈
Y and all x ∈ τ(Z)\Y,
[yMx if and only if y'Mx] and [xMy if and only if xMy'].

(2.15.3) For all y,y' ∈
Y and all MC ∈ majoritycycles(τ(Z),R)
such that MC∩Y={y}, substitute(min(MC),y,y')
= min(substitute(MC,y,y')).

Condition 2.15.1 restricts similar sets to sets
containing at least two of the alternatives in Z, an obvious requirement.
Condition 2.15.2 embodies our basic intuition for broadening of clone sets; it
restricts similar sets by requiring that for every alternative x outside a
similar set, the M and ~ aggregate relations between x and every similar
alternative are the same. (The restriction has been relaxed here by
ignoring relations outside the top cycle, since our purpose is to establish the
robustness of ICA by demonstrating the existence of voting procedures which
satisfy as strong an independence criterion as possible.) As
mentioned earlier, 2.15.2 does not restrict similarity enough to allow existence
of voting schemes which are independent from similar alternatives, which is why
condition 2.15.3 is imposed. Condition 2.15.3 restricts similar sets
further by requiring that additional aggregate properties of R, the
smallest majorities in every majority cycle containing exactly one of the
similar alternatives, are the same when any similar alternative is substituted
in those majority cycles. (Like 2.15.2, 2.15.3 has been relaxed by
ignoring cycles outside the top cycle in order to strengthen the independence
criterion as much as possible to establish the robustness of ICA.)

It is easy to see that every clone set of (Z,R) is a similar set of (Z,R).
The alternatives in a clone set remain similar under small perturbations of R,
since similarity is based on aggregate properties of R which are
insensitive to small perturbations.

The following statements hold for all non-empty Z ⊆ A:

(2.15.4) If #Z > 1, then Z is (trivially)
a similar set of (Z,R).

(2.15.5) If #τ( Z,R)
> 1, then τ( Z,R) is a similar set of (Z,R).

(2.15.6) If #(Z\τ( Z,R))
> 1, then Z\τ( Z,R) is a similar set of (Z,R).

The following definition, which provides a measure
of the relative "clonishness" of two
subsets of alternatives, is
needed for a mild technical restriction in ISA, analogous to
the innocuous
technical restriction in ICA, to avoid a problem in some rare scenarios
involving indeterminism:

Definition (2.16): Given A,
R, Z ⊆ A, Y ⊆
Z, and X ⊆ Z such that X\Y ≠
∅, let clonescore(Y,X,R) denote maximum(over
x ∈ X\Y) of #{i ∈
N such that there exists y,y' ∈ Y such that
[xPiy and not xPiy'] or [yPix and not y'Pix]}.
Given A, R, X ⊆ A, and Y ⊆
A such that X\Y ≠ ∅
and Y\X ≠ ∅, X is more
clone-like than Y with respect to R if and only if clonescore(X,Y,R)
< clonescore(Y,X,R).

Note that if Y is a clone set of (Z,R),
it follows immediately from condition 2.6.2 of the definition of clone set that clonescore(Y,X,R)
= 0 for all X ⊆ Z such that X\Y ≠
∅). Thus, no subset of alternatives is more clone-like than
a clone set.

We can now formally state the ISA criterion:

Criterion
(2.17): Independence from Similar Alternatives
(ISA):
A voting scheme f is independent from similar alternatives if and only if
fx(Z,R) = fx(Z\Y',RZ\Y') for all A, all R, all non-empty Z
⊆
A, all Y such that Y is a similar set of (Z,R), all Y' ⊂
Y,
and all x ∈ A\Y such that [for all X ⊆
Z\Y such that x ∈ X and all non-empty Ys
⊂ Y such that X∪Ys
is a similar set of (Z,R), Y is more clone-like than X∪Ys
with respect to R].

Lemma: Satisfaction of ISA
implies satisfaction of ICA.

Proof: Suppose Y is a clone
set of (Z,R) and f is a voting scheme which satisfies ISA.
Since every clone set of (Z,R) is a similar set of (Z,R),
satisfaction of ISA implies that fx(Z,R) = fx(Z\Y',RZ\Y')
for all Y' ⊂ Y
and all x ∈ A\Y such that [for all X ⊆
Z\Y such that x ∈ X and all non-empty Ys ⊂
Y such that X∪Ys is a similar set of (Z,R),
Y is more clone-like than X∪Ys with
respect to R]. To demonstrate satisfaction of ICA, we need to show
that fx(Z,R) = fx(Z\Y',RZ\Y')
for all Y' ⊂ Y
and all x ∈ A\Y such that [there exists no y
∈ Y such that xIiy for all i ∈
N]. Since for all x ∉ Z, fx(Z,R)
= 0 and fx(Z\Y',RZ\Y') = 0, we
need only establish the following statement:

For all x ∈
Z\Y such that [there exists no y ∈ Y such that xIiy
for all i ∈ N] and all X ⊆
Z\Y such that x ∈ X and all non-empty Ys
⊂ Y such that X∪Ys
is a similar set of (Z,R), Y is more clone-like than X∪Ys
with respect to R.

Assume x ∈
Z\Y and there exists no y ∈ Y such that xIiy
for all i ∈ N and X ⊆
Z\Y and x ∈ X and Ys is a non-empty
strict subset of Y and X∪Ys is a similar
set of (Z,R). We need to establish that Y is more clone-like than X∪Ys
with respect to R.

Since x ∈
Z\Y and x ∈ X, (X∪Ys)\Y
≠ ∅. Also, since X ⊆
Z\Y and Ys ⊆ Z, X∪Ys
⊆ Z. Also, since Y is a clone set of (Z,R),
Y ⊆ Z. Given the conclusions of the last
three statements, clonescore(Y,X∪Ys,R)
is defined, and since Y is a clone set, clonescore(Y,X∪Ys,R)
= 0.

There are two cases to consider:

Case 1: clonescore(X∪Ys,Y,R)
> 0. In this case, it is obvious that Y is more clone-like than X∪Ys
with respect to R.

Case 2: clonescore(X∪Ys,Y,R)
= 0. We want to show that this case contradicts the assumptions.

So the only x ∈
A\Y for which fx(Z,R) need not equal fx(Z\Y',RZ\Y')
are those x for which there exists an X∪Ys
containing x such that X∪Ys is a similar
set and clonescore(X∪Ys,Y,R)
= 0.

Since the technical restriction in
ISA ("for all X ⊆ Z\Y such that x ∈
X and all non-empty Ys ⊂ Y such that X∪Ys
is a similar set of (Z,R), Y is more clone-like than X∪Ys")
is not identical to the technical restriction in ICA ("there is no y ∈
Y such that {x,y} is an exact clone set"), it may not be obvious that ISA
is a stronger criterion than
ICA. But this is indeed the case, since when Y is a clone set, the
restriction condition in ICA holds if the restriction condition in ISA
holds. Suppose that Y is a clone set and x ∉ Y
and {x,y'} is an exact clone set for some y' ∈ Y;
then Y\{y'} is not more clone-like than {x,y'} because

maximum(over x ∈
X\Y) of #{i ∈ N such that [xPiy
for all y ∈ Y] or [yPix for all y ∈
Y]} is less than maximum(over y ∈ Y\X) of
#{i ∈ N such that [yPix for
all x ∈ X] or [xPiy for all x ∈
X]}

#{i ∈
N such that [xPiy for all y ∈
Y\{y'}] or [yPix for all y ∈ Y\{y'}]} =
0. Therefore, since every clone set is also a similar set,
satisfaction of ISA implies satisfaction of ICA and IECA, and failure
of IECA or ICA implies failure of ISA.

There are other ways similar sets could have been
defined (e.g., by replacing x#y with x#y-y#x and replacing z#z' with z#z'-z'#z in the definition of min(MC,R))
which would make it possible to satisfy some variant of ISA but
not the ISA defined here. So it could be controversial to insist upon satisfaction of
the ISA defined here. But the
purpose of ISA is primarily to strengthen the argument for requiring satisfaction of
ICA by showing that manipulation resistance is robust.

The following properties of similar (clone, exact
clone) sets and majority cycles will be useful when proving subsequent lemmas
and theorems:

Lemma (2.18): (Splittability of
majority cycle) If Y is a similar set of (Z,R) and MC ∈
majoritycycles(Z,R)
and #(MC∩Y)>1 and (x,x') ∈
min(MC,R)
and Y∩{x,x'}≠{x,x'},
then there exists an MC1 ∈ majoritycycles(Z,R) such that
#(MC1∩Y)=1 and (x,x') ∈
min(MC1,R). (Proof in
Appendix A1.)

Lemma 2.18 states that any majority
which is not a pair
of similar alternatives and which is
smallest in
a majority cycle containing at least one similar alternative is also
smallest in a majority cycle containing exactly one similar alternative.

Lemma (2.19): (Substitutability of similar
alternative) If Y is a similar set of (Z,R) and y,y' ∈
Y and MC ∈ majoritycycles(Z,R) and MC∩Y={y}, then
substitute(MC,y,y') ∈ majoritycycles(Z,R).
(Proof in appendix A1.)

Lemma 2.19 states that for any set
of alternatives which is a majority cycle and contains exactly one alternative
of a similar set, the set obtained by substituting any similar alternative is
also a majority cycle. This is important in 2.15.3 of the definition of
similar set, which uses the expression min(substitute(MC,y,y')),
since the definition of min() requires that its argument is a majority
cycle.

Lemma (2.20): For any sets Y, Z, and Zs such
that
Y is a similar (clone, exact clone) set of (Z,R) and Zs∩τ( Z,R)
≠ Ø
and #(Y∩Zs)>1, Y∩Zs is a similar (clone, exact clone)
set of (Z∩Zs,R). (Proof
in appendix A1.)

Lemma 2.20 essentially states that
when Y is a similar set of Z, Y∩Zs is a
similar set of Z∩Zs, assuming a couple of
obvious restrictions. The restriction about
the top cycle could be dropped and the lemma would still hold for
clone sets and exact clone sets. Also, without the restriction about the
top cycle, the lemma would hold for similar sets if the definition of similarity were narrowed by
replacing "τ(Z)" with "Z"
everywhere in the definition of similar set. But we prefer to define similar sets
broadly in order to make ISA stronger.

3. Correspondences and
Tie-breaking

We seek voting schemes which satisfy ISA (and thus ICA and IECA). Such voting
schemes are
considered to be "completely" independent from
similar alternatives. In particular, we seek such voting schemes which are
Condorcet-consistent. It is useful to first describe
related properties involving correspondences, and to define some
notation denoting the breaking of ties of a correspondence by
applying other correspondences and/or a voting scheme.

Definition (3.1): A correspondence
is a function g : 2A\Ø xRn → 2A\Ø
which satisfies the following conditions:

(3.1.1) g(Z,R)
⊆ Z for all A, all R, and
all non-empty Z ⊆ A.

(3.1.2) g(Z,R')
= g(Z,R) for all A, all R, all non-empty Z ⊆ A,
and all R' such that R' = R.

(3.1.3) g(Z,RZ)
= g(Z,R) for all A, all R, and all non-empty Z ⊆ A.

A correspondence takes as its arguments a non-empty
subset of the alternatives and a profile of voters' rankings and returns a
non-empty subset of the alternatives. Condition 3.1.1 means no alternative
outside Z will be chosen. Condition 3.1.2 means the set of chosen
alternatives depends only on the voters' rankings. One implication is that
correspondences are deterministic, leaving nothing to chance. Condition
3.1.3 means that a correspondence neglects alternatives outside the feasible set
Z; the set of
chosen alternatives is independent of neglected alternatives.

A correspondence already mentioned is the top cycle
τ(Z,R).

Definition (3.2): A correspondence g
is partially independent from similar (clone, exact clone) alternatives if and only if
the following conditions hold for
all A, all R, all non-empty Z ⊆ A,
all Y such that Y is a similar
(clone, exact
clone) set of (Z,R), and all Ys ⊂
Y:

(3.2.1) g(Z,R)
∩ A\Y = g(Z\Ys,RZ\Ys) ∩
A\Y

(3.2.2) g(Z,R)
∩ Y = Ø if and only if g(Z\Ys,RZ\Ys)
∩ Y = Ø

Condition 3.2.1 requires that for all non-similar
(non-clone, non exact clone) alternatives x (i.e., x ∈
Z\Y),
x is chosen when no alternatives in Y are neglected if and only if x is chosen when any
strict subset of Y is neglected.
Condition 3.2.2 requires that at least one alternative in Y is chosen when no
alternatives in Y are neglected if and only if at least one alternative in Y is
chosen when any strict subset of Y is neglected.

A correspondence which is partially independent
and which usually selects a singleton in some applications,
(e.g., large public elections) can be considered "practically independent" in those
applications since a tie-breaker would rarely be required. However, to
demonstrate complete independence we must consider those cases where the chosen
set contains more than one alternative, where a tie-breaker would be needed.

Definition (3.3): Let
g//f
denote the function defined with respect to two functions g
and f, where g is a correspondence and f is either a
correspondence or a voting scheme, which first applies g
and then applies f to the subset chosen by g. Formally,

(g//f)(Z,R) =
f(g(Z,R),R).

Depending on whether f is a correspondence
or a voting scheme, g//f is a correspondence or a voting
scheme. The
notation can be extended to g1//g2//g3//...//f
where g1, g2, g3, ... are
correspondences and f is either a correspondence or a voting scheme.
Left-to-right precedence is implied; for instance g1//g2//f
= (g1//g2)//f.

We also define the notation [g]
to indicate that a correspondence g is repeated until convergence; i.e.,
until further repetition would eliminate no more alternatives. To illustrate the meaning of [g] we could abuse
notation by writing
[g] = g//g//g//g//g//... Instead
we define [g] formally as follows:

Definition (3.4): Let [g] denote the
correspondence defined inductively with respect to a correspondence g
by the following rules:

(3.4.1)
For all positive integers r, let gr be the
correspondence defined
inductively with respect to a correspondence g by the two rules:

(3.4.3)
Abbreviating r' = r'(g,Z,R),
[g] is the correspondence such that [g](Z,R) = gr'(Z,R)
for all A, all R, and all
non-empty Z ⊆ A.

Lemma (3.5): r'(g,Z,R)
< #Z for all correspondences g, all A, all R, and all
non-empty Z ⊆ A.
(Proof in appendix A1.)

The // and [] notation is credited
to Bruce Anderson, whose definitions are slightly different in that they
informally stipulate that preferences regarding eliminated alternatives are
neglected. In our context, where voting schemes and correspondences are
required to neglect alternatives not in their feasible subset of alternatives,
it is not necessary to add that stipulation.

The Condorcet-consistent ISA-consistent
voting scheme whose existence is established in the following sections takes the form
Ψ//Dissimilar(Ψ), where Ψ is a correspondence which
satisfies the top cycle criterion and is
partially independent from similar alternatives and Dissimilar(Ψ) is a
voting scheme which satisfies ISA
and may involve some randomness.

Criterion (3.6): Top Cycle
(TC). A correspondence g satisfies top cycle if and
only if g(Z,R) ⊆ τ(Z,R) for all
A, all R, and all non-empty Z ⊆
A. A voting scheme f
satisfies top cycle if and
only if fx(Z,R) = 0 for all
A, all R, all non-empty Z ⊆
A,
and all x ∉ τ(Z,R).

Lemma (3.7): If g1 and
g2 are correspondences which satisfy top cycle and are partially independent from similar alternatives,
then g1//g2
is a correspondence which satisfies top
cycle and is partially independent from similar
alternatives. (Proof
in appendix A1.)

Lemma (3.8): If g is a correspondence
which satisfies top cycle and is partially independent from similar alternatives, then [g] is a
correspondence which satisfies top cycle and is partially independent from similar
alternatives. (Proof in
appendix A1.)

Lemma (3.9): If g is a correspondence
which satisfies top cycle and is
partially independent from similar alternatives, and f is a voting scheme which satisfies ISA, then g//f and [g]//f are
voting schemes which satisfy top cycle and ISA. (Proof
in appendix A1.)

4. The Ψβ>1
and Ψβ> correspondences

In this section we define a correspondence which satisfies
top cycle, is partially independent from similar (clone, exact clone)
alternatives, and which would rarely need a tie-breaker in large public
elections. First we define a useful subset of the alternatives, obtained
by ignoring the smallest majority (or majorities) of every majority cycle and
selecting the alternatives which are unbeaten pairwise in the set of non-ignored
majorities:

Definition (4.1):
Given A, R,
and non-empty Z ⊆ A, let
ψ1(Z,R) denote the set of
alternatives {x ∈ Z | for all z ∈
Z such that zMx, there exists at least one MC ∈
majoritycycles(Z,R)
such that (z,x) ∈ min(MC,R)}.

Note the similarity between the
definition of ψ1 and the definition
of τ. The clause beginning with "for
all z ∈ Z such that zMx" would simply
embody the majority rule heuristic if the
"there exists..." condition
were replaced by the false condition. Condorcet's paradox tells
us that
the majority rule heuristic alone would be
too strong since it could result in the
elimination of every
alternative. Relaxing the heuristic when there exists a majority cycle
MC
such that "(z,x) ∈ min(MC,R)" guarantees
ψ1 will
never be empty: Since due to cycles
it is not always possible to honor all
majority opinions,
a remedy
is to ignore the smallest
majority of every majority cycle.
The
non-ignored majorities will always be transitive
so ψ1
cannot be empty. Appendix A2 provides an example
illustrating the calculation of ψ1.

Definition (4.2): Ψ1
and Ψβ>

(4.2.1) Ψ1
is the correspondence defined by the rule Ψ1(Z,R) =
ψ1(Z,R) for all A, all R,
and all non-empty Z ⊆
A.

(4.2.2) Ψ
is the correspondence [Ψ1].

In 1785 the Marquis
de Condorcet of the French Academy of Sciences proposed a voting
correspondence that has been interpreted by H
Peyton Young as being similar to Ψ1.
According to Young, when there is
no Condorcet winner "... one successively deletes
from that opinion the
propositions that have the least plurality, and one adopts the
opinion from
those that remain." Or, as translated by Ian
McLean, "... then we shall
successively exclude from it the
propositions with the lowest plurality." Taken literally,
this procedure is equivalent to a non-iterative minimax
since it chooses the alternative
whose largest pairwise opposition is
smallest. Unfortunately, Condorcet's writing was
terse and ambiguous. For one thing, it is not clear how he measured the
size of each
proposition's plurality. The word
"plurality" is often interpreted now-a-days as the margin
of
difference between two counts, but French
dictionaries of his time defined plurality as
the larger of the counts. (See
the Dictionnaire de L'Académie Française, 1694: "plus grande
quantité,
plus grand nombre.") On
the other hand, Condorcet estimated the probability
that a jury reaches the correct decision as a function of the margin of difference between
two counts.
McLean pointed out some of these ambiguities: Translating from p. lxvii
of
Condorcet's "Essai": "Now imagine that the 3 propositions with
plurality support
form one of the 2 contradictory systems..." which suggests Condorcet
intended
supporting coalition and not margin of
difference. McLean later writes:

"Where does this leave
Condorcet's choice rule? - still ambiguous, because
the choice rule consistent
with the ordering rule ("select the candidate with the
maximum likelihood
of being the correct one") is not the same as the choice rule
embodying
the Condorcet criterion ("select the Condorcet winner if there is
one").
We aim to show that Condorcet never resolved the tension
between probability
and social choice in his writings on elections during the
last nine years of his life."
[McLean 1995,
p.22]

Due to considerations of voting strategies and
equilibria, which Condorcet apparently did
not consider, we prefer to measure
size by using the supporting coalition (the larger of the
two counts) rather than
the margin of difference. (See the definition of min()
in section 2).
Condorcet explicitly assumed all votes
would be made in good faith, and assuming also
complete strict preferences, then, when
those assumptions hold, neglecting strategic effects,
both measures
would result in the choice of the same alternative(s).

It is unclear whether Condorcet
was aware that the wording of his proposal, taken literally,
could elect a
Condorcet loser (an alternatives that is less preferred by a majority in each
of
its pairings). His wording does not discriminate between cyclic
propositions, such as those
in the top cycle, from acyclic propositions, such as
the defeats of alternatives outside the
top cycle by the alternatives in the top
cycle. Since there is no need to discard acyclic
propositions when
resolving cycles, it is easy to imagine that Condorcet would have
been
comfortable with one or both of the following procedures:

"Successively delete the
smallest of the cyclic propositions
until no cycles remain."

"Successively delete the
smallest of the cyclic propositions,
recalculating cyclicity after each
deletion, until no cycles remain."

Measuring the sizes of
propositions by their supporting coalitions, the first of these two is Ψ1.
The latter, which we will call Ψ2, elects
a subset of
ψ1, is nearly as easy to calculate as Ψ1,
and satisfies ICA (and probably ISA too, though this has not been verified) when
accompanied
by a suitable tie-breaker. The difference between Ψ1
and Ψ2 is that Ψ2
recalculates cyclicity
but Ψ1 does
not. Ψ also recalculates cyclicity, in a sense,
but less often than Ψ2. This
difference can occasionally result in different winners. An argument for
preferring Ψ over
[Ψ2]
is that when they choose different winners, the winner
according to Ψ tends to beat
pairwise the winner
according to [Ψ2] more often than vice
versa, according to computer
simulations using randomly generated
rankings. Since we are concerned here only with
establishing the existence
of a voting scheme which satisfies ISA, we do not consider Ψ2
further in this document, apart from a brief comment below about its
decisiveness.

Theorem (4.4): Ψ1
and Ψ are partially
independent from similar (clone, exact clone)
alternatives. (The proof
is in appendix A1.)

Like the top cycle correspondence τ, Ψ1 is too indecisive to be
practical by itself. Even
with many
voters, its indecisiveness increases asymptotically with the
number of alternatives,
according to computer simulations
using randomly generated preference orders. However,
the decisiveness of Ψ is
comparable to other correspondences which are prominent in the
literature,
leaving little to chance, particularly in large public elections. Ψ2
is more decisive
than Ψ1 and would be
reasonably decisive in large public elections. [Ψ2]
is slightly more
decisive than Ψ2, and its
decisiveness is comparable to other prominent correspondences.

To explore the issue of "practical"
independence, which requires both partial independence
and
"reasonable" decisiveness, computer simulations using randomly
generated orders of
preference were run to quantify the decisiveness of Ψ1
and Ψ. A portion of the decisiveness
measurements are
summarized in appendix A3. The results show that the
decisiveness of Ψ1
and Ψ
increases as the number of voters increases, and the decisiveness of Ψ is comparable
to the decisiveness of
Borda given many voters. The results suggest that Ψ would be
reasonably decisive in large public elections, meaning that a poor choice of
tie-breaker
(e.g., "random") for Ψ would not substantially affect its properties in those elections.

5. The Dissimilar(g)
voting scheme

If we were merely interested in satisfaction of ICA
and IECA, not in ISA, a voting scheme
which is a variation of Random Dictator would be a suitable tie-breaker for
correspondences
which are partially independent from similar alternatives, clone
alternatives, and/or exact clone
alternatives. Random Dictator would
suffice if voters' rankings are required to be strict
orderings, but allowing
indifference adds a degree of complexity. Random Voter Hierarchy,
a generalization of Random Dictator, chooses one
alternative from a feasible subset by
randomly picking a ballot and eliminating
all but that ballot's top-ranked feasible alternative(s), repeating until either
one alternative remains or all ballots have been randomly
picked.
In the latter case, which can occur only if all voters report
indifference between the
remaining alternatives, one of the remaining alternatives is randomly chosen.

Let random(S)
denote the function which, given any non-empty set S, randomly
chooses one element of
S, giving an equal 1/#S chance to each element of S.
(When S is a singleton, random(S) selects the lone element of
S with certainty.)

For all i
∈ N, all A, all R
and all Z ⊆
A, let top(i,Z,R)
denote the subset of
alternatives {x ∈ Z such that, for all
y
∈ Z, [xPiy
or xIiy]}.

Random
Voter Hierarchy and RandomVoterHierarchyScheme:

Let RandomVoterHierarchy(Z,R)
denote the probabilistic function
defined for all A, all R and
all non-empty Z ⊆
A by this procedure:

RandomVoterHierarchyScheme denotes
the voting scheme such that, for all A,
all R, all non-empty Z ⊆
A, and all x ∈ A,
RandomVoterHierarchySchemex(Z,R)
equals the probability that x = RandomVoterHierarchy(Z,R).

It would be desirable to also have
a formulaic definition of RandomVoterHierarchyScheme,
but that would require
additional complicated notation and would probably not increase
its ease of
understanding.

Given a correspondence g which is partially
independent from similar (or clone) alternatives,
the voting scheme g//RandomVoterHierarchyScheme
satisfies IECA and ICA. (We do
not attempt to prove this here since we are
interested in ISA.) Satisfaction of ISA is more
demanding, however, and
requires a more sophisticated voting scheme as tie-breaker.

Dissimilar(g) is a voting scheme
defined with respect to a voting correspondence g.
Dissimilar(g)(Z,R)
eliminates all but one alternative from each identifiable
similar set
of (Z,R) by iteratively applying g, and if necessary random(), to each
such set.
We formally define Dissimilar(g) and describe its properties:

Given a correspondence g,
Dissimilar(g)(Z,R)
is defined for all A, all R
and all finite non-empty Z ⊆
A by this procedure:

Step 1. Let
L be a #A-tuple of real numbers having one component for each alternative in A,
and initally let Lx = 1 for all x ∈
Z and let
Lx = 0
for all x ∈ A\Z. Let Represent
be a #A-tuple of alternatives having one component for each alternative in A,
and initially let Represent(x) = x
for all x ∈
A.
Let D be a subset of the alternatives, and initially let D = Z.

Step 2. Repeat while
#D > 1:

(2.1) Pick a set
Y such that Y is a similar set of (D,R)
and no strict subset
of Y is a similar set of (D,R).

In step 2.1, at least one such
similar set exists while #D>1 since D itself is trivially a similar set of (D,R).
An algorithm to identify such a similar set which is fast even when Z is large
is yet to be found, however, suggesting that Dissimilar is likely to be
practical only as a tie-breaker.

There is an
ambiguity in step 2.1 since there may be more than one similar set meeting the
specified constraint. If the similar sets meeting the constraint are
disjoint, the ambiguity is innocuous since the choice between them will not
ultimately make a difference. However, if two or more of them are not
disjoint, the choice between them can make a difference. Though this
problem does not affect partial independence or practical independence, a claim
of complete independence from similar alternatives cannot be sustained unless
this ambiguity is resolvable by finding a way to narrow the definition of
similarity or independence that does not sacrifice manipulation
resistance. Still under development, such a narrowing is expected to
involve the identification of the "most clone-like" of the
non-disjoint sets and rejecting the similarity of the "less
clone-like." Letting Yu denote the union of the
non-disjoint "near-similar" sets, each set Y might be scored according
to the largest number of voters who ranked an alternative in Yu\Y
strictly over or under every alternative in Y, and calling "similar"
only the set having the largest score (a maximaximax function).

Step 2.6 eliminates from
D all of Y except an alternative which is "g-best" of Y.
Since Y is not a singleton, step 2.6 must eliminate at least one alternative
from D, so the number of times step 2 is performed must be less than #Z.)

In step 3, each component of L is a real number in
[0,1], the components of L sum to 1, Lx=0 for all x ∉
Z,
etc. So for any correspondence g, Dissimilar(g)
is a voting scheme:

Lemma (5.5): If g is a
correspondence which satisfies top cycle,
then Dissimilar(g) satisfies top cycle. (Proof
in appendix A1.)

Theorem (5.6): If g is a correspondence which
satisfies top cycle and is
partially independent from similar alternatives, then Dissimilar(g) satisfies
top cycle, ISA, ICA, and IECA. (Proof
in appendix A1.)

6. Conclusions

Ψ//Dissimilar(Ψ) is a
single-winner voting
function which satisfies ISA, ICA, IECA, top cycle, anonymity, neutrality, and
other desirable criteria. Satisfaction of ISA suggests it is less
manipulable than many voting procedures (a conclusion which will be bolstered by
a consideration of certain strategy criteria not discussed here).

Dissimilar(Ψ) is also a single-winner
voting
function which satisfies ISA, ICA, IECA, top cycle, etc. In a sense,
Dissimilar(Ψ) is simpler than Ψ//Dissimilar(Ψ), but
Ψ
is reasonably decisive so in practice Ψ//Dissimilar(Ψ) would be
much simpler to tally. If voting procedures need to be defined as simply as
possible in order to gain public acceptance, it would be reasonable to simply
propose Ψ and not specify the tie-breaker, since Ψ is
"practically" independent from similar alternatives in public
elections

Appendix A1: Proofs of Lemmas and
Theorems

Lemma (2.14): (Rotations of majority cycle) If mc = {z1,z2,...,zk}
(k>2) is a majority cycle of (Z,R), then mc' = {z2,...,zk,z1}
is a majority cycle of (Z,R), and the majority sequence associated with mc
is identical to the majority sequence associated with mc'.

Lemma (2.15): (Splittability of
majority cycle) If Y is a similar set of (Z,R), and z and z' are
two alternatives in Z such that zMz' and at least one of z and z' is in
Z\Y, and mc is a majority cycle of (Z,R)
such that z and z' are either adjacent in mc or last and first in mc,
z#z' is last in the majority sequence associated with mc, and at least
two distinct alternatives in mc are in Y, then there exists a majority
cycle of (Z,R) (call it mc1 without loss of generality) such that
z and z' are adjacent in mc1, z#z' is last in the majority
sequence associated with mc1, and exactly one alternative in mc1
is in Y.

Proof: Assume the
premise holds. By induction on lemma 2.14 (rotatability), there must be a
majority cycle mc' = {z1,z2,...,zk}
obtained by rotation of mc such that z1=z and z2=z' and z#z' is last in the majority sequence associated with mc'.
Since mc' contains at least two distinct alternatives in Y and at least
one alternative in Z\Y, mc and mc' must contain at least two
distinct alternatives in Z\Y, else 2.12.2 of the definition
of similar set would be contradicted. So k > 4. Again
by 2.12.2, at least two distinct alternatives in Z\Y must be either adjacent in mc
or last and first in mc and either adjacent in mc' or last and
first in mc'. (In other words, by 2.12.2, in no majority cycle can
two alternatives in Y be separated by exactly one alternative not in Y).

Of all the alternatives which are
in both mc' and Y, let l be the position in mc' of the one
positioned lowest in mc', let s be the position in mc' of
the one positioned second lowest in mc', and let h be the position
in mc' of the one positioned highest in mc'. (Possibly h = s
since there may be as few as two alternatives which are in both mc' and
Y.)
Thus zl, zs, zh ∈
Y. Since at least one of z1 and z2 are in Z\Y, 3<s<h<k.
Since there are at least two alternatives in mc' which are in Y, l<s
and l<h. By 2.12.2 of the definition of similar set and
by the definition of majority cycle, the following statements hold:

(2.15.1) If l >
2, then zl-1M zh.

(2.15.2) If h <
k, then zlM zh+1.

(2.15.3) If l = 2
and h = k,
then zlM z1.

(2.15.4) If l = 1,
then zs-1 M zl.

There are three cases to consider:

Case 1: l >
2. By 2.15.1, the ordered set mc1 = {z1,z2,...,zl-2,zl-1,zh,zh+1,...,zk}is
a majority cycle of (Z,R). Since z1=z and
z2=z', z and z' are adjacent in mc1.
Exactly one alternative (zh) in Y is in mc1.
By the definition of majority
sequence, z#z' is last in the majority sequence associated with mc1.

Case 2: l =
2. By 2.15.2 and 2.15.3, the ordered set mc1
= {z1,z2,zh+1,...,zk} is a
majority cycle of (Z,R). Since z1=z and z2=z',
z and z' are adjacent in mc1. Exactly one alternative
(z2) in Y is in mc1. By the definition of
majority sequence, z#z' is last in the majority sequence associated
with mc1.

Case 3: l =
1. By 2.15.4, the ordered set mc1 = {z1,z2,...zs-1}is
a majority cycle of (Z,R). Since z1=z and z2=z',
z and z' are adjacent in mc1. Exactly one alternative
(z1) in Y is in mc1. By the definition of
majority sequence, z#z' is last in the majority sequence associated
with mc1.

In all cases there exists a
majority cycle meeting the desired conditions, so the lemma is proved.

Lemma
(2.16): (Substitutability of similar alternative) If Y is a similar set of
(Z,R) and mc = {z1,z2,...zk)
(k>2) is a majority cycle of (Z,R) such that z1 ∈ Y and
z1 is the only alternative in mc which is in Y, then for all y∈Y {y,z2,...,zk} is a majority cycle of (Z,R).

Proof: By 2.12.2 of
the definition of similar set, for all y∈ Y, z1Mz2
implies yMz2 and zkMz1
implies zkMy. By the definition of
majority cycle, {y,z2,...,zk} must be a majority
cycle of (Z,R).

Lemma
(2.17): For any sets Y, Z, and Zs such that Y is a similar
(clone, exact clone) set of (Z,R) and Zs ⊆
Z
and at least one alternative in Zs is in the top cycle of (Z,R)
and Y∩Zs ≠ Ø,
Y∩Zs is a similar (clone, exact
clone) set of (Zs,R).

Proof: Abbreviate Ys
= Y∩Zs and assume the premise, considering first the case where Y is
a similar set of (Z,R). We show that all the
similarity conditions of 2.12 hold for Ys with respect to (Zs,R):

Since Ys ⊆
Zs and Ys ≠ Ø, 2.12.1 holds for Ys with
respect to (Zs,R).

Since Ys ⊆
Y and Zs\Ys ⊆
Z\Y and 2.12.2 holds for Y with respect to (Z,R), it must be the case
that for all y,y' ∈ Ys and all x ∈
Zs\Ys,
yMx if and only if y'Mx, and xMy
if and only if xMy'. So 2.12.2 holds for Ys
with respect to (Zs,R).

Since Zs ⊆
Z and Zs ≠ Ø, every majority cycle
of (Zs,R) is a majority cycle of (Z,R), and their
associated majority sequences
are invariant since R is held fixed. Since at least one
alternative in Zs is in τ(Z,R), τ(Zs,R)
is a non-empty subset of τ(Z,R), and every X = {x1,x2,...,xk}
which is a subset of τ(Zs,R)\Ys
is also a subset of τ(Z,R)\Y. Given
these facts, and since all four conditions of 2.12.3 hold for Y with respect to
(Z,R), it must be the case that for all y,y' ∈
Ys
and all X = {x1,x2,...,xk} such that X ⊆
τ(Zs,R)\Ys and both mc1={y1,x1,x2,...,xk}
and
mc2={y2,x1,x2,...,xk}
are majority cycles of (Zs,R), all four conditions of 2.12.3
hold for Ys with respect to (Zs,R).

Since all conditions of 2.12 hold
for Ys with respect to (Zs,R), the lemma is proved
for when Y is a similar set of (Z,R). Since every clone set of (Z,R)
is a similar set of (Z,R) and every exact clone set of (Z,R) is a
clone set of (Z,R), the lemma also holds for when Y is a clone set of (Z,R)
and for when Y is an exact clone set of (Z,R).

Lemma
(2.18): (Disjointness of
similar sets) If Y and Y' are similar sets of (Z,R) and neither is
a subset of the other, then Y∩Y' = Ø. FALSE!!

Proof: Suppose
not. Then there is at least one alternative y in Y but not in Y', and
there is at least one alternative y' in Y' but not in Y. By the definition
of similar set, the following statements...

Lemma (3.5):r'(g,Z,R)
< #Z for all correspondences g, all A, all R, and any
non-empty Z ⊆ A.

Proof: Since Z
and R are held fixed, abuse notation by abbreviating. Also
abbreviate #g = #(g(Z,R)). Since gr
is a correspondence, for all r∈{1,2,...} 0 < #gr
< #Z. Since r' is the smallest integer r>1 such that gr
= gr+1, r' is also the smallest integer r>1 such that #gr
= #gr+1. So #gr > #gr+1
for all r∈{1,2,..,r'-1}, and since #gr' > 0,
by induction #gr'-k >
k for all k∈{0,1,...,r'-1}.

Suppose to the contrary that r' > #Z. Then #Z
∈ {1,2,...,r'-1} and when k = #Z, #gr'-k>
#Z. But this would contradict #gr < #Z for
all r in {1,2,...}, so r' > #Z must be false. Thus r'
< #Z and the lemma is proved.

Lemma
(3.7): If g1 and
g2 are correspondences which satisfy top cycle and are partially
independent from similar
alternatives,
then g1//g2
is a correspondence which satisfies top cycle and is partially independent from similar
alternatives.

Proof: Since R is
fixed, abbreviate g(...) for g(...,R). Assume the
premise of the
lemma. Let Z be a non-empty subset of A and let Y be a similar set of (Z,R) and let Ys be
a strict subset of Y. We first establish that g1//g2
is partially independent from
similar alternatives by showing that conditions 3.2.1 and 3.2.2 hold for g = g1//g2.

Since g1 is partially independent from similar alternatives,
the following two statements must hold:

(3.7.1) g1(Z)
∩ A\Y = g1(Z\Ys)
∩ A\Y

(3.7.2) g1(Z)
∩ Y = Ø if and only if g1(Z\Ys)
∩ Y = Ø

By properties of set intersection and union, the following two statements must hold:

Case 3: g1(Z)∩Y
≠ Ø and g1(Z) ∩
A\Y ≠ Ø. Since
g1 satisfies top cycle, g1(Z) is a
non-empty subset of τ(Z,R). By lemma
2.17 (letting Zs be g1(Z)∪Y), Y is a similar set
of (g1(Z)∪Y,R).
Letting Ys1 = Y \ (g1(Z)∩Y) implies Ys1
⊂ Y. Since g2 is partially
independent from similar alternatives, and since (by properties of intersection
and union) g1(Z) = (g1(Z)∪Y)\Ys1,
by 3.2.1 and 3.2.2 the following two statements must hold in case 3:

(3.7.5) g2(g1(Z)∪Y) ∩ A\Y = g2(g1(Z))
∩ A\Y

(3.7.6) g2(g1(Z)∪Y) ∩ Y = Ø if and only if g2(g1(Z))
∩ Y = Ø

Letting Ys2 = Y \
(g1(Z\Ys)∩Y) implies Ys2
⊂ Y. By properties of intersection and union, (g1(Z\Ys)∪Y)\Ys2
= g1(Z\Ys). By 3.7.1, g1(Z\Ys)∪Y
= g1(Z)∪Y. Combining
the last two equations gives (g1(Z)∪Y)\Ys2
= g1(Z\Ys). Since g2 is partially
independent from similar alternatives, by 3.2.1 and 3.2.2 the following two
statements must hold in case 3:

(3.7.7) g2(g1(Z)∪Y) ∩ A\Y = g2(g1(Z\Ys))
∩ A\Y

(3.7.8) g2(g1(Z)∪Y) ∩ Y = Ø if and only if g2(g1(Z\Ys))
∩ Y = Ø

The left side of 3.7.5 is
identical to the left side of 3.7.7, and the left side of 3.7.6 is identical to
the left side of 3.7.8. So both conditions 3.2.1 and 3.2.2 hold for g1//g2
in case 3.

The similarity conditions hold in all three cases, establishing
that g1//g2 is partially independent from
similar alternatives.

Since g1
satisfies top cycle, g1 must choose a non-empty subset of the
top cycle, and g1//g2 must choose a
non-empty subset of what g1 chooses. So g1//g2
must choose a non-empty subset of the top cycle, which establishes that g1//g2
satisfies top cycle.

Lemma
(3.8): If g is a correspondence
which satisfies top cycle and is partially independent from similar alternatives, then [g] is a
correspondence which satisfies top cycle and is
partially independent from similar
alternatives.

Lemma
(3.9): If g is a correspondence
which satisfies top cycle and is
partially independent from similar alternatives and f is a single-winner voting function which satisfies ISA, then g//f and [g]//f are
single-winner voting functions which satisfy top cycle and ISA.

Proof: The proof for g//f
closely resembles the proof of lemma 3.7. Then by lemma 3.8, the proof for [g]//f
follows immediately.

Lemma (4.3):
Ψ1 and Ψ
satisfy the top cycle criterion.

Proof: By the definition
of the top cycle, yMx for all y ∈ τ(Z,R)
and all x ∉ τ(Z,R). So condition 4.1.1
of the definition of ψ1
holds for all y ∈ τ(Z,R) and all x
∉ τ(Z,R).
By the definition of majority cycle,
there are no majority cycles of (Z,R) containing both x and y, for any y ∈
τ(Z,R)
and any x ∉ τ(Z,R). So condition 4.1.2 holds for all
y ∈ τ(Z,R) and all x
∉ τ(Z,R).
Since τ(Z,R) is a non-empty subset of Z, for all x
∉ τ(Z,R) there exists at least one y
∈ τ(Z,R)
for which both 4.1.1 and 4.1.2 hold. Therefore the ψ1(Z,R)
set includes no alternatives which are not in τ(Z,R). By the
definition of Ψ1, Ψ1(Z,R)
= ψ1(Z,R) so Ψ1 satisfies top
cycle. By induction and the same reasoning in lemma 3.7 which showed that g1//g2
satisfies top cycle if g1 satisfies top cycle, Ψ
satisfies top cycle because Ψ = [Ψ1].

Proof: Since R
is held fixed, we abuse notation by abbreviating (...,R) as (...).
First we show that Ψ1
is partially independent from similar alternatives: Assume Z is a
non-empty subset of A, Y is a similar
set of (Z,R) and Ys is a strict subset of
Y. By the definition of Ψ1, Ψ1(Z)
= ψ1(Z) and Ψ1(Z\Ys)
= ψ1(Z\Ys), so we need to
establish the following two statements:

(4.4.1) ψ1(Z)
∩ A\Y = ψ1(Z\Ys)
∩ A\Y

(4.4.2) ψ1(Z)
∩ Y = Ø if and only if ψ1(Z\Ys)
∩ Y = Ø

By the definition of ψ1,
x ∈ ψ1(Z) if and only if x∈Z and there is no z∈Z for which both of the
following conditions hold:

(4.4.3)
zMx.

(4.4.4)
For all mc such that mc is a
majority cycle of (Z,R) and mc contains both z and x, z#x is not last in
ms(mc).

By the definition of ψ1,
x ∈ ψ1(Z\Ys) if and only if
x ∈ Z\Ys and there is no z' ∈
Z\Ys for which both of the
following conditions hold:

(4.4.5)
z'Mx.

(4.4.6)
For all mc' such that mc' is a
majority cycle of (Z\Ys,R) and mc' contains both z' and
x, z'#x is not last in ms(mc').

To establish 4.4.1 there are four cases to
consider:

Case 4.4.1.1: x ∈ A\Y
and x ∈ ψ1(Z). We must show
that x ∈ ψ1(Z\Ys).
By the definition of ψ1, x∈Z,
and since x ∈ A\Y, x ∈
Z\Y and x ∈ Z\Ys. By the definition of ψ1
there is no z∈Z for which both 4.4.3 and 4.4.4 hold. By the
definition of majority cycle, every majority cycle of (Z\Ys,R)
is a majority cycle of (Z,R), so there cannot be any z' ∈
Z\Ys
such that 4.4.5 and 4.4.6 both hold. Therefore, by the definition of ψ1,
x ∈ ψ1(Z\Ys).

Case 4.4.1.2: x ∈
A\Y
and x ∈ ψ1(Z\Ys).
We must show that x ∈ ψ1(Z).
This follows by reasoning similar to that in case 4.4.1.1, plus the use 2.12.2
of the definition of similarity, plus lemma
2.15 (splittability) and lemma 2.16
(substitutability) and 2.12.2 and 2.12.3 of the definition
of similarity to show that the majority cycles of (Z,R) which are
not majority cycles of (Z\Ys,R) are either redundant or
irrelevant for determination of whether x ∈ ψ1(Z).

Case 4.4.1.3: x ∈
A\Y
and x ∉ ψ1(Z). We must
show that x ∉ ψ1(Z\Ys).
By the definition of ψ1, x∈ Z,
so x ∈ Z\Y, and x ∈ Z\Ys. By the definition of ψ1
there is a z∈Z for which both 4.4.3 and 4.4.4 hold. There are two
subcases to consider:

Case 4.4.1.3.1: z ∈
Z\Y. Since Ys ⊂ Y, z ∈
Z\Ys.
By the definition of majority cycle, every majority cycle of (Z\Ys,R)
is a majority cycle of (Z,R), and since z#x is not last in the
majority sequence associated with any majority cycle of (Z,R) which
contains z and x, z#x is not last in the majority sequence associated with
any majority cycle of (Z\Ys,R) which contains z and
x. Therefore, by the definition of ψ1,
x ∉ ψ1(Z\Ys).

Case 4.4.1.3.2: z ∈
Y. By 4.4.3 and similarity, yMx for all y∈ Y. By 4.4.4
and similarity, for all y∈Y there is no majority cycle mc of (Z,R)
such that mc contains both y and x and y#x is last in ms(mc). Since Ys
⊂ Y, yMx for all y ∈
Y\Ys. Since all majority cycles
of (Z\Ys,R) are majority cycles of (Z,R), for all y
∈ Y\Ys there is no majority cycle mc' of (Z\Ys,R)
such that mc' contains both y and x and y#x is last in ms(mc'). Therefore, by the definition of ψ1,
x ∉ ψ1(Z\Ys).

Case 4.4.1.4: x ∈
A\Y
and x ∉ ψ1(Z\Ys).
We must show that x ∉ ψ1(Z).
There are two subcases to consider:

Case 4.4.1.4.1: z ∈
Z\Y. The reasoning in this case is to case 4.4.1.3.1 as the reasoning
in case 4.4.1.2 is to case 4.4.1.1.

Case 4.4.1.4.2: z ∈
Y. The reasoning in this case is to case 4.4.1.3.2 as the reasoning in
case 4.4.1.2 is to case 4.4.1.1.

So 4.4.1 has been established in all cases.
To establish 4.4.2 there are four cases to consider:

Case 4.4.2.1: ψ1(Z)∩Y
≠ Ø. We must show that ψ1(Z\Ys)∩Y
≠ Ø. There exists an alternative in Y, call it y0 with
no loss of generality, such that y0 ∈ ψ1(Z)∩Y.
By the definition of ψ1, for all z∈Z
such that zMy0, there exists a majority cycle mc of (Z,R)
such that z#y0 is last in ms(mc). There are four cases
to consider:

Case 4.4.2.1.1: z ∈
Z\Y. By lemma 2.15 (splittability) and lemma 2.14 (rotatability), for
all z ∈ Z\Y such that zMy0 there exists a majority cycle mc'
of (Z,R) such that mc' = {z,y0,...,zk}
and y0 is the only alternative in Y which is in mc' and
z#y0 is last in ms(mc'). By lemma 2.16
(substitutability), for all y∈Y and for all z ∈
Z\Y such that zMy0,
there is a majority cycle mc" of (Z,R) such that mc"
= {z,y,...,zk} and y is the only alternative in Y which is in mc"
and z#y is last in ms(mc"). Furthermore, mc"
is also a majority cycle of (Z\Ys,R). Therefore, for
all y ∈ Y\Ys and all z ∈
Z\Y such that zMy, there exists
a majority cycle of (Z\Ys,R) containing z and y such that
z#y is last in its associated majority sequence. NEED TO SHOW THAT
NOT ALL Ys ARE KNOCKED OUT BY Ys.

Case 4.4.2.1.2: z ∈
Y\Ys and #(Y\Ys) > 1. y0
is in Y\Ys?

Case 4.4.2.1.3: z ∈
Y\Ys and #(Y\Ys) = 1. y0 is
in Y\Ys?

Case 4.4.2.1.4: z ∈
Ys. y0 ∈ Y\Ys?

Case 4.4.2.2: ψ1(Z\Ys)∩Y
≠ Ø. We must show that ψ1(Z)∩Y
≠ Ø.

Case 4.4.2.3: ψ1(Z)∩Y
= Ø. We must show that ψ1(Z\Ys)∩Y
= Ø.

Case 4.4.2.4: ψ1(Z\Ys)∩Y
= Ø. We must show that ψ1(Z)∩Y = Ø.

Having established 4.4.1 and 4.4.2 in all cases, Ψ1
is partially independent from similar alternatives. Since every clone set
of (Z,R) is a similar set of (Z,R) and since every exact clone set
of (Z,R) is a clone set of (Z,R), Ψ1
is partially independent from clone alternatives and Ψ1
is partially independent from exact clone alternatives. By lemma 3.8 and
by induction on lemma 4.3, Ψ is partially independent
from similar alternatives. Since every clone set of (Z,R) is a
similar set of (Z,R) and since every exact clone set of (Z,R) is a
clone set of (Z,R), Ψ is partially independent
from clone alternatives and Ψ is partially
independent from exact clone alternatives.

Lemma (5.3): If g is a
correspondence which satisfies top cycle,
then Dissimilar(g) satisfies top cycle.

Proof:

Theorem (5.4): If g is a correspondence which
satisfies top cycle and is
partially independent from similar alternatives, then Dissimilar(g) satisfies
top cycle, ISA, ICA, and IECA.

c ∉ ψ1: Letting x=c, condition 4.1.1
does not hold for
z=b and
condition 4.1.2 does not hold for z=b since (b,c)∉min({a,b,c}).

Therefore, ψ1
= {a,b}, and thus Ψ1({a,b,c},R)
= {a,b}.

To calculate Ψ,
repeat Ψ1 until convergence: (Ψ1//Ψ1)({a,b,c},R))
=Ψ1({a,b},R)
= {a} (since aMb and c is neglected). So in this
example, a repetition of Ψ1
suffices to eliminate all but one alternative, and Ψ({a,b,c},R)
= {a}.

Appendix
A3: Decisiveness of Ψ1 and Ψ

Computer simulations using randomly
generated orders of preference were run in an attempt to quantify the decisiveness of
Ψ1
and Ψ. For comparison the decisiveness of Borda was also
tallied. The number of voters and the number of alternatives were varied,
and for each set of parameters the computer generated 1,000 scenarios of random
preference orders and tallied each scenario by all three procedures. The
three columns in the middle of the tables show the decisiveness of the
procedures.

The two columns at the right of the tables
provide an additional comparison showing the overall majority preferences
regarding the Ψ winner and the Borda winner: The "Ψ M
Borda" column shows the percentage of scenarios where both procedures were
decisive and more voters ranked the Ψ winner over the Borda winner than
vice versa. The "Borda M Ψ" column shows the percentage
of scenarios where both procedures were decisive and more voters ranked the
Borda winner over the Ψ winner than vice versa. (The two numbers in
each row don't total 100% because sometimes both procedures chose the same
alternative and sometimes one or both procedures were indecisive.)

All the scenarios in table A-3.1 involved strict
preferences only. In all scenarios in table A-3.2, the preference relation
between every adjacent pair of alternatives in every voter's ranking had a 0.3 probability of being
indifference instead of strict preference.

Table A-3.1 Decisiveness given strict
preferences

Voters
Alternatives Ψ1Ψ
Borda Ψ M Borda
Borda MΨ

100
3 89.7% 90.8%
96.9%
7.5% 0.0%

100
6 81.7% 88.8%
97.8%
16.6% 1.4%

100
12 65.2% 86.6%
97.2%
24.2% 3.8%

1000
3 99.2% 99.4%
99.9%
11.0% 0.0%

1000
6 93.4% 99.4%
99.1%
20.2% 2.8%

1000
12 80.3% 98.8%
99.5%
28.3% 5.6%

Table A-3.2 Decisiveness given non-strict
preferences

Voters
Alternatives Ψ1Ψ
Borda Ψ M
Borda
Borda MΨ

100
3 93.0% 94.3%
97.0%
6.8% 0.5%

100
6 85.7% 92.2%
98.5%
16.6% 3.1%

100
12 71.3% 91.6%
98.9%
23.4% 5.6%

1000
3 99.1% 99.3%
99.7%
9.5% 0.8%

1000
6 94.1% 99.4%
99.6%
20.3% 3.4%

1000
12 79.5% 98.9%
100.0%*
27.8% 6.8%

* A 100% decisiveness value does not imply
absolute decisiveness. A more accurate test involving many thousands of scenarios
would produce a value below 100%.

The tables show that the
decisiveness of all three procedures increased as the number of voters
increased, and the decisiveness of Ψ is comparable to the decisiveness of
Borda given 1,000 voters. The tables suggest that Ψ would be
reasonably decisive in large public elections, meaning that a poor choice of
tie-breaker for Ψ would not substantially affect its properties in those elections. The assumption regarding non-strictness of
reported preferences does not appear to significantly affect
decisiveness.

Appendix
A4: Example calculation of Dissimilar

This example involves three
alternatives {a,b,c} such that {b,c} is a similar
set and is nearly a clone set.
All three alternatives tie pairwise, so a tie-breaker is needed to break a
three-way tie if a pairwise correspondence is employed. We assume the
voting procedure satisfies anonymity and neutrality, so a would have a ½
chance of being elected if a has only one opponent. To satisfy
independence from similar alternatives, a must still have a ½ chance of
being elected when both b and c compete.

All three pairings of
the alternatives are 50:50 ties. Alternatives b and c are
nearly clones: by slightly changing two ballots to produce the 25,25,25,25
scenario, b and c would technically be clones since every voter
would rank a either over both b and c or under both b
and c. As is, the RandomHierarchyScheme
tie-breaker (or RandomDictator) gives a a 49/100 chance of being elected
since 49 voters rank a best and 51 voters rank a worse than
best. We show here that Dissimilar(g)
gives a a ½ chance, assuming g is a pairwise correspondence
satisfying anonymity and neutrality (meaning that, given a feasible set of two
tied alternatives, g chooses both alternatives).

In step 2.1,
Dissimilar finds {b,c} is (Oops NOT!) the only similar
set meeting the requirements. (Though {a,b,c}is a
similar set, so is a strict subset of {a,b,c}.) So
Y={b,c}.

In step 2.2, g({b,c})={b,c}
by assumption, so G={b,c}. Then x' is set to either b
or c, at random. Here we assume x'=b. (The reader
may verify that the lottery will be the same if x'=c.)

In step 2.3,
nothing happens since Y\G is empty.

In step 2.4, Lb
and Lc are divided by #G=2, so L=(1,½,½). Also, Representc
is set to b so Represent=(a,b,b).

In step 2.5, c
is eliminated from D so D={a,b}.

Since #D=2 > 1,
step 2 is repeated:

In step 2.1,
Dissimilar finds {a,b} is the only similar set meeting the
requirements. So Y={a,b}.

In step 2.2, g({a,b})={a,b}
by assumption, so G={a,b}. Then x' is set to either a
or b, at random. Here we assume x'=a. (The reader
may verify that the lottery will be the same if x'=b.)

In step 2.3,
nothing happens since Y\G is empty.

In step 2.4, La,
Lb and Lc are divided by #G=2, so
L=(½,¼,¼). Also, Representb and Representc
are set to a so Represent=(a,a,a).

In step 2.5, b
is eliminated from D so D={a}.

Since #D=1, step 2
does not repeat again.

In step 3, Dissimilar
returns the lottery (½,¼,¼). This means that independence from clone
alternatives is robust over small perturbations of the ballots which make the
clones premise not strictly hold.